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Mirrors > Home > MPE Home > Th. List > o1resb | Unicode version |
Description: The restriction of a function to an unbounded-above interval is eventually bounded iff the original is eventually bounded. (Contributed by Mario Carneiro, 9-Apr-2016.) |
Ref | Expression |
---|---|
rlimresb.1 | |
rlimresb.2 | |
rlimresb.3 |
Ref | Expression |
---|---|
o1resb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | o1res 13383 | . 2 | |
2 | rlimresb.1 | . . . . . . 7 | |
3 | 2 | feqmptd 5926 | . . . . . 6 |
4 | 3 | reseq1d 5277 | . . . . 5 |
5 | resmpt3 5329 | . . . . 5 | |
6 | 4, 5 | syl6eq 2514 | . . . 4 |
7 | 6 | eleq1d 2526 | . . 3 |
8 | inss1 3717 | . . . . . 6 | |
9 | rlimresb.2 | . . . . . 6 | |
10 | 8, 9 | syl5ss 3514 | . . . . 5 |
11 | 8 | sseli 3499 | . . . . . 6 |
12 | ffvelrn 6029 | . . . . . 6 | |
13 | 2, 11, 12 | syl2an 477 | . . . . 5 |
14 | 10, 13 | elo1mpt 13357 | . . . 4 |
15 | elin 3686 | . . . . . . . . . 10 | |
16 | 15 | imbi1i 325 | . . . . . . . . 9 |
17 | impexp 446 | . . . . . . . . 9 | |
18 | 16, 17 | bitri 249 | . . . . . . . 8 |
19 | impexp 446 | . . . . . . . . . 10 | |
20 | rlimresb.3 | . . . . . . . . . . . . . . 15 | |
21 | 20 | ad2antrr 725 | . . . . . . . . . . . . . 14 |
22 | 9 | adantr 465 | . . . . . . . . . . . . . . 15 |
23 | 22 | sselda 3503 | . . . . . . . . . . . . . 14 |
24 | elicopnf 11649 | . . . . . . . . . . . . . . 15 | |
25 | 24 | baibd 909 | . . . . . . . . . . . . . 14 |
26 | 21, 23, 25 | syl2anc 661 | . . . . . . . . . . . . 13 |
27 | 26 | anbi1d 704 | . . . . . . . . . . . 12 |
28 | simplrl 761 | . . . . . . . . . . . . 13 | |
29 | maxle 11420 | . . . . . . . . . . . . 13 | |
30 | 21, 28, 23, 29 | syl3anc 1228 | . . . . . . . . . . . 12 |
31 | 27, 30 | bitr4d 256 | . . . . . . . . . . 11 |
32 | 31 | imbi1d 317 | . . . . . . . . . 10 |
33 | 19, 32 | syl5bbr 259 | . . . . . . . . 9 |
34 | 33 | pm5.74da 687 | . . . . . . . 8 |
35 | 18, 34 | syl5bb 257 | . . . . . . 7 |
36 | 35 | ralbidv2 2892 | . . . . . 6 |
37 | 2 | adantr 465 | . . . . . . 7 |
38 | simprl 756 | . . . . . . . 8 | |
39 | 20 | adantr 465 | . . . . . . . 8 |
40 | 38, 39 | ifcld 3984 | . . . . . . 7 |
41 | simprr 757 | . . . . . . 7 | |
42 | elo12r 13351 | . . . . . . . 8 | |
43 | 42 | 3expia 1198 | . . . . . . 7 |
44 | 37, 22, 40, 41, 43 | syl22anc 1229 | . . . . . 6 |
45 | 36, 44 | sylbid 215 | . . . . 5 |
46 | 45 | rexlimdvva 2956 | . . . 4 |
47 | 14, 46 | sylbid 215 | . . 3 |
48 | 7, 47 | sylbid 215 | . 2 |
49 | 1, 48 | impbid2 204 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 <-> wb 184
/\ wa 369 e. wcel 1818 A. wral 2807
E. wrex 2808 i^i cin 3474 C_ wss 3475
if cif 3941 class class class wbr 4452
e. cmpt 4510 |` cres 5006 --> wf 5589
` cfv 5593 (class class class)co 6296
cc 9511 cr 9512 cpnf 9646 cle 9650 cico 11560
cabs 13067 co1 13309 |
This theorem is referenced by: chpo1ub 23665 dchrisum0lem2a 23702 pntrsumo1 23750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-cnex 9569 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-mulcom 9577 ax-addass 9578 ax-mulass 9579 ax-distr 9580 ax-i2m1 9581 ax-1ne0 9582 ax-1rid 9583 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 ax-pre-ltadd 9589 ax-pre-mulgt0 9590 ax-pre-sup 9591 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-riota 6257 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6701 df-2nd 6801 df-recs 7061 df-rdg 7095 df-er 7330 df-pm 7442 df-en 7537 df-dom 7538 df-sdom 7539 df-sup 7921 df-pnf 9651 df-mnf 9652 df-xr 9653 df-ltxr 9654 df-le 9655 df-sub 9830 df-neg 9831 df-div 10232 df-nn 10562 df-2 10619 df-3 10620 df-n0 10821 df-z 10890 df-uz 11111 df-rp 11250 df-ico 11564 df-seq 12108 df-exp 12167 df-cj 12932 df-re 12933 df-im 12934 df-sqrt 13068 df-abs 13069 df-o1 13313 df-lo1 13314 |
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